Nilradical

An infinite subgroup with spread exactly one

Kourovka Notebook · 21.38Answer: AffirmativeNilradical v0

The result

Theorem

The subgroup of Thompson's group F consisting of maps with equal slopes at the two endpoints is infinite and has spread exactly one: every nonidentity element has a generating companion, but some two elements share none.

Scope of the contribution

The subgroup consists of maps with equal endpoint slopes; the spread condition holds exactly for k ≤ 1. The key generation argument specializes Gili Golan-Polak’s published theorem.

Specialization and assembly of prior arguments, proof and formalization

Mathematical dependencies and credit

Specialization, assembly of the generation and fixed-point arguments, proof and formalization by Nilradical v0. The generation theorem is due to Gili Golan-Polak; related fixed-point arguments are due to Golan-Polak and Bleak–Donoven–Harper–Hyde.

  • Nilradical v0 — discovery, proof and formalization
  • C. Donoven and S. Harper — original question
  • Gili Golan-Polak — prior generation theorem
  • Group-approximation contributors — adapted formal foundations

Proof source

The source and theorem endpoints refer to the pinned revision.

  • Kourovka.P21_38.notebookStatement

Source revision
5a6b2c18e326b7b0281f00b64cade629acbfe1f5

Proof sourceProof overview

Recorded checks and review

Recorded proof checks

Strict verification passed on 16 September 2026 for 7 endpoints: fresh compilation, frozen-statement comparison, a three-axiom policy, and checks by both Lean and Nanoda. Independent statement review passed. Final human statement acceptance was recorded on 2026-09-16 for the linked proof snapshot.

Recorded proof checks record

Statement review

Recorded comparison of the formal statement and its intended mathematical scope.

Statement review record

Human statement acceptance

Human statement acceptance for this pinned source snapshot. This is separate from refereeing of the exposition or a priority determination.

Human statement acceptance record

Historical literature review

A dated, bounded literature search. This historical review is not a certificate of absolute priority; current related work is listed separately.

Historical literature review record

Mathematical note

Agent-generated exposition; not refereed, revised . The note does not change the proof statement or its recorded checks.

Read the mathematical note

Cite this contribution

Nilradical v0. An infinite subgroup with spread exactly one. Cite the original mathematics and the formalization you use, retaining the source credits above.

BibTeX citation
@misc{nilradical_kourovka_21_38,
  author = {{Nilradical}},
  title = {An infinite subgroup with spread exactly one},
  url = {https://github.com/alunik/kourovka-lean/blob/5a6b2c18e326b7b0281f00b64cade629acbfe1f5/Kourovka/Problems/P21_38/README.md},
  note = {Source revision 5a6b2c18e326b7b0281f00b64cade629acbfe1f5}
}

Using this contribution does not require adding Nilradical or anyone from the project as a coauthor.